Let $X(i)$ be a simple symmetric random walk in $\mathbb{Z}$ starting from the origin. Let $\tau$ be a stopping time for the random walk, namely for any $k \in \mathbb{N}$, the event $\{\tau=k\}$ depends only on the first $k$ steps of the random walk. An example of $\tau$ might be the first time the simple random walk hits a given vertex $m \in \mathbb{Z}$.
Assume we know the expectation $E[\tau]$ and that $E[ \, X(\tau) \, ] = 0$. Is there a way to infer some information on $ E[ \, | X(\tau) | \, ] $, the expected distance of the random walk from the origin at time $\tau$? In other words, are $E[\tau]$ and $E[ \, \, | X(\tau) | \, \, ]$ connected by some inequality?